A subclass of paranormal including class of log-hyponormal and several related classes (Operator Inequalities and related topics)

A subclass of paranormal including class of log-hyponormal

and

several

related

classes

東京理科大理 山崎丈明 (Takeaki Yamazaki)

東京理科大理 伊藤公訴 (Masatoshi Ito)

東京理科大理 古田孝之 (Takayuki

Furuta)

Abstract

$..\mathrm{T}$his report is based on the following preprint:

T.Furuta, M.Ito and T.Yamazaki, A subclass _ofparanormal operators including

class

_of

$log$-hyponormal and several related classes, to appear in Scientiae

Mathe-maticae.

We shall introduce a new class “class $\mathrm{A}$” given by an operator inequality which

includes the class of$\log$-hyponormal operators and is included in the class of

para-normal operators. It turns out that our results contain another proof of Ando’s

result [3] which every $log$-hyponormal operator is paranormal. Moreover we shall

introduce new classes related to class A operators and paranormal operators.

1

Introduction

A capital letter means a bounded linear operator on a complex Hilbert space $H$. An

operator $T$ is said to be positive (denoted by _{$T\geq 0$}) if $(Tx, x)\geq 0$ for all $x\in H.$and

also an operator $T$ is said to be strictly positive (denoted by $T>0$) if $T$ is positive and

invertible.

An operator $T$ is said to be _$p$-hyponormal if $(T^{*}T)^{p}\geq(TT^{*})^{p}$ for a positive number

$p$ and $log$-hyponormal if $T$ is invertible and $\log T^{*}T\geq\log TT*.$ p–Hyponormal and

log-hyponormal operators were defined as extensions of log-hyponormal one, i.e., $T^{*}T\geq TT^{*}$, and also they have been studied by many authors, for instance, $[1][2][5][8][13][14][17][18]$

and [19]. By the celebrated L\"owner-Heinz inequality $ttA\geq B\geq 0$ ensures $A^{\alpha}\geq B^{\alpha}$

for

any $\alpha\in[0,1]"$, every p–hyponormal operator is $q$-hyponormal for $p\geq q>0$. And every

invertible $p \frac{-}{}\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}1$ operator is $\log$-hyponormal since $\log t$ is an operator monotone

function.

An operator $T$ is paranomal if $||T^{2_{X}}||\geq||TX||^{2}$ for every unit vector $x\in H$. It has

been studied by many authors, so there are too many to cite their

references.’

for instance,

$[3][9]$ and [15]. Ando [3] proved the following result.

Theorem A.l ([3]). Every $log$-hyponormal operator is paranormal.

$\mathrm{h}$ this paper, firstly we shall introduce a new class “class $\mathrm{A}$” given $\dot{\mathrm{b}}\mathrm{y}$

an operator inequality which properly includes the class of $\log$-hyponormal operators and is properly included in the class of paranormal operators. It turns out thatour results containanother

proofofTheoremA.l. Secondly we shall introduce$\mathrm{n}e\mathrm{w}$ classesrelatedto class A operators

and paranormal operators. Finally we shall give several examples to show that inclusion

2A subclass of paranormal

operators

including

class

of

log-hyponormal

We shall introduce a new class of operators as follows:

Definition 1.

.An

operator $T$ belongs to class $A$

if

$|T^{2}|\geq|T|^{2}$

.

(2.1)

We would like to remark that class “_$A$” _{is named after the ‘iabsolute” values of two}

operators $|T^{2}|$ and $|T|$ in (2.1). We call an operator $T$class A operator briefly if$T$ belongs to class A. We obtain the following results on class A operators.

Theorem 1.

(1) Every $log$-hyponormal operator is class $A$ operator.

(2) Every class $A$ operator is paranormal operator.

Theorem 1 contains another proofofTheorem A. 1. Thefollowing theorems and lemma

play an important role in the proofs ofthe results in this paper.

Theorem B.l $([6][10])$

.

Let$A$ and$B$ be positive invertible operators. Then the following

properties are mutually equivalent:

(i) $\log A\geq\log B$

.

(ii) $A^{p}\geq(A^{E}2B^{p}A^{\epsilon}2)^{\frac{1}{2}}$

for

all _{$p\geq 0$}

.

(iii) $A^{f} \geq(A^{\frac{r}{2}}B^{p}A\frac{r}{2})^{\frac{r}{p+r}}$

for

all_{$p\geq 0$} and $r\geq 0$.

We remark that equivalence relation between (i) and (ii) is shown in [4].

Theorem B.2 ($\mathrm{H}\ddot{\mathrm{o}}1\mathrm{d}\mathrm{e}\Gamma^{-}\mathrm{M}\mathrm{c}\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{y}$ inequality [16]). Let $A$ be a positive operator.

Then the following inequalities hold

for

all $x\in H$:

(i) $(A^{\mathrm{r}}x,x)\leq(Ax, x)’||X||\mathrm{z}(1-r)$

for

_{$0<r\leq 1$}

.

(ii) $(A^{r_{X}}, x)\geq(Ax, x)’||X||^{2}\langle 1-r$)

for

$r\geq 1$.

As a slightly modification of [11, Lemma 1], we have the following lemma. Lemma B.3. Let $A$ and $B$ be invertible operators. Then

$(BAA^{*}B*)^{\lambda}=BA(A^{*}B^{*}BA)^{\lambda}-1A*B^{*}$

Proof of

Theorem 1.

Proof

of

(1). Suppose that $T$ is $\log$-hyponormal. $T$ is $\log$-hyponormal iff

$\log|T|^{2}\geq\log|T^{*}|^{2}$. (2.2)

By the equivalence between (i) and (ii) of Theorem B.l, (2.2) is equivalent to

$|T|\mathrm{t}2p\geq(|T|^{\mathrm{p}}.|\tau*|2p|T|^{p})^{\frac{1}{2}}$ for all$p\geq 0$. (2.3)

Put $p=1$ in (2.3), then we have

$|T|^{2}\geq(|\tau||\tau*|^{2}|T|)^{\frac{1}{2}}$. (2.4)

By Lemma B.3 and $|T^{*}|^{2}=TT^{*},$ $(2.4)$ holds iff

$|T|^{2} \geq|T|T(T^{*}|\tau|2\tau)\frac{-1}{2}T^{*}|T|$

iff

$(T^{*}|\tau|^{2}T)^{\frac{1}{2}}\geq^{\tau*}T$, (2.5)

so that

$|T^{2}|\geq|T|^{2}$,

that is, $T$ is class A.

Proof of

(2). Suppose that $T$ is class $\mathrm{A}$, i.e., $l$

$|T^{2}|\geq|T|^{2}$

.

(2.1)

Then for every unit vector $x\in H$,

$||\tau^{2_{X}}||2=((\tau 2)*\tau^{2_{X,x}})$

$=(|\tau^{2}|^{2_{X}}, X)$

$\geq(|T^{2}|x, x)^{2}$ by (ii) of Theorem B.2

$\geq(|T|^{2}x, x)2$ by (2.1)

$=||T_{X1}|^{4}$

Hence we have

$||T^{2_{X}}||\geq||TX||^{2}$ for every unit vector $x\in H$

,

so that $T$ is paranormal.

3Several

classes

related

to

class

A and

paranormal

In this section, we shall discuss extensions of class A operators and paranormal

oper-ators. We shall introduce new classes of operators as follows:

Definition 2.

(1) For each $k>0_{y}$ an operator$T$ is class $A(k)$

if

$(T^{*}|\tau|^{2k}\tau)^{\frac{1}{k+1}}\geq|T|^{2}$. (3.1)

(2) For each $k>0$, an operator$T$ is _{$absolute- k-pa\Gamma anormal$}

if

$|||T|^{k}\tau_{X||}\geq||TX||k+1$

for

every unit vector$x\in H$. (3.2)

An operator $T$ is class A (resp. paranormal) if and only if $T$ is class $\mathrm{A}(1)$ (resp.

$\mathrm{a}\mathrm{b}\mathrm{S}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}-1-\mathrm{P}^{\mathrm{a}}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}1)$

.

Now we shall discuss the inclusion relations among these classes.

Theorem 2.

(1) Every invertible class $A$ operator is class _$A(k)$ operator

for

$k\geq 1$

.

(2) Every paranormal operator is $ab_{SO}lute- k- pa\Gamma anormal$ operator

for

$k\geq 1$.

(3) For each $k>0_{J}$ every class $A(k)$ operator is absolute-k-paranormal operator.

We need the following theorem in order to give a proof of Theorem 2.

Theorem C.l ([12]). Let $A$ and $B$ be positive invertible operators such that

$A^{\beta_{0}}\geq(A^{\rho_{2}\rho_{2}\ovalbox{\tt\small REJECT}}B^{\alpha \mathrm{O}}\mathrm{n}\Delta A)^{\overline{\alpha}}0+\overline{\beta 0}$holds

for fixed

$\alpha_{0}>0$ and$\beta_{0}>0$

.

Then the following inequality holds:

$A^{\beta}\geq(A^{\mathrm{g}}2B\alpha A^{e}2)\overline{\alpha}+E_{\overline{\rho}}$

for

all _{$\alpha\geq\alpha_{0}$} and $\beta\geq\beta_{0}$

.

Proof of

Theo-rem

2.

Proof of

(1). Suppose that $T$ is class $\mathrm{A}$, i.e.,

$|T^{2}|\geq|T|^{2}$. (2.1)

(2.1) holds if and only if

$(T^{*}|T|^{2}\tau)^{\frac{1}{2}}\geq T^{*}T$

.

(2.5)

By Lemma $\mathrm{B}.3,$ $(2.5)$ holds iff

$T^{*}|T|(|T| \tau\tau^{*}|\tau|)\frac{-1}{2}|T|\tau\geq T^{*}T$

iff

Applying Theorem C.l to (2.4), we have

$|T|^{2k} \geq(|T|^{k}|T^{*}|2|\tau|^{k})\frac{k}{k+1}$ for _{$k\geq 1$}. (3.3)

By Lemma B.3 and $|T^{*}|^{2}=TT^{*},$ $(3.3)$ holds iff

$|T|^{2k} \underline{>}|T|k\tau(\tau*|T|^{2}kT)\frac{-1}{k+1}\tau^{*}|\tau|k$ for _{$k\geq 1$}

iff

$(T^{*}|T|^{2k}\tau)^{\frac{1}{k+1}}\geq|T|^{2}$ for $k\geq 1$,

so that $T$ is class $\mathrm{A}(k)$ for $k\geq 1$.

Proof

of

(2). Suppose that $T$ is paranormal. Then for everyunit vector _{$x\in H$} and $k\geq 1$,

$|||T|^{k}T_{X1}|^{2}=(|T|^{2}k\tau X, TX)$

$\geq(|T|^{2}\tau_{X}, \tau x)k||TX||^{2(1-k})$ by (ii) of Theorem B.2

$=||T^{2}X||2k||\tau X||2\langle 1-k)$

$\geq||TX||4k||T_{X1}|2(1-k)$ by paranormality of$T$

$=||TX||2\langle k+1)$.

Hence we have

$|||T|^{k}\tau X||\geq||TX||k+1$ for every unit vector $x\in H$ and $k\geq 1$,

so that $T$ is $\mathrm{a}\mathrm{b}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}e- k_{\mathrm{P}}- \mathrm{a}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}1$for $k\geq 1$.

Proof

of

(3). Suppose that $T$ is class $\mathrm{A}(k)$ for $k>0$, i.e.,

$(\tau^{*}|\tau|^{2k}\tau)^{\frac{1}{k+1}}\geq|T|^{2}$ for $k>0$. (3.1)

Then for every unit vector $x\in H$,

$|||T|^{k}T_{X}||^{2}=(T^{*}|T|2k\tau x, x)$

$\geq((T^{*}|T|^{2}kT)\frac{1}{k+1}x, x)^{k+1}$ by (ii) of Theorem B.2

$\geq(|T|^{2}x, x)k+1$ by (3.1)

$=||Tx||2(k+1)$.

Hence we have

$|||\tau|^{k}\tau_{x}||\geq||TX||k+1$ for every unit vector $x\in H$,

so that $T$ is $\mathrm{a}\mathrm{b}_{\mathrm{S}}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}- k$-paranormal for $k>0$.

Whence the proof ofTheorem 2 is complete. $\square$

Theorem 3. Let $T$ be an invertible class $A(k)$ operator

for

$k>0$. Then

$f(l)=(T^{*}|\tau|2lT)^{\frac{1}{l+1}}$

is increasing

_for

$l\geq k>0$, and the following inequality holds:

$f(l)\geq|T|^{2},$ $i.e.,$ $T$ is class $A(l)$

for

$l\geq k>0$

.

Theorem 4. Let $T$ be an $ab_{S}olute-k$-paranormal operator

for

$k>0$. Then

for

every unit

vector $x\in H$,

$F(l)=|||T|^{l}T_{X}|| \frac{1}{l+1}$

is increasing

_for

$l\geq k>0$, and the following inequality holds:

$F(l)\geq||TX||$, $i.e_{J}.T$ is $ab_{S}olute- l_{- pa}ranormal$

for

$l\geq k>0$.

Theorem 3 states the following: An operatorfunction$f(l)$ assertsthat every class $\mathrm{A}(k)$

operator is class $\mathrm{A}(l)$ for $l\geq k>0$

.

Similarly, Theorem4 states the following: A function

ofnorm $F(l)$ asserts that every absolute-k-paranormal operator is absolute-l-paranormal

for $l\geq k>0$.

In order to give a proof of Theorem 3, we need the following theorem which is an

extension of Theorem C.1.

Theorem C.2 ([12]). Let $A$ and $B$ be positive invertible operators such that

$A^{\beta_{0}}\geq(A^{\underline{\rho}_{\mathit{1}}}2B^{\alpha 0}A^{-}2)^{\frac{\beta_{0}}{\alpha_{0+\rho_{0}}}}\beta_{\mathrm{L}}$ holds

for

fixed

$\alpha_{0}>0$ and $\beta_{0}>0$. Then

for fixed

$\delta\geq-\beta_{\mathit{0}_{J}}$

$f( \alpha,\beta)=A\frac{-\beta}{2}(A2B\alpha A2)E\mathrm{g}\frac{\delta+\beta}{a+\beta}A^{\frac{-\mathcal{B}}{2}}$

is a decreasing

_{function of}

both $\alpha$ and $\beta$

for

$\alpha\geq\max\{\delta, \alpha_{0}\}$ and$\beta\geq\beta_{0}$

.

Proof of

Theorem 3. Suppose that $T$ is class $\mathrm{A}(k)$ for $k>0$, i.e.,

$f(k)=(\tau^{*}|\tau|^{2k}\tau)^{\frac{1}{k+1}}\geq|T|^{2}$. (3.1)

By Lemma B.3, (3.1) holds iff

$T^{*}|T|^{k}(|\tau|kT\tau*|\tau|^{k})^{\frac{-k}{k+1}|T|}k\tau\geq T^{*}T$

iff

By applying Theorem C.2 to (3.4),

$g(l)=|\tau|^{-}l(|T|l|T^{*}|^{2}|T|l)^{\frac{l}{l+1}}|T|^{-l}$

is decreasing for $l\geq k>0$

.

And we have

$g(l)=|T|^{-}l(|T| \iota|T*|^{2}|\tau|l)\frac{l}{l+1}|\tau|-\iota$

$=|T|^{-}l(|T|^{\iota l}\tau T^{*}|\tau|)^{\frac{l}{l+1}1}\tau|^{-^{\iota}}$

$=T(T^{*}|T|^{2l}T)^{\frac{-1}{l+1}}\tau^{*}$ by Lemma B.3

$=T \{(T^{*}|T|^{2}lT)\frac{1}{l+1}\}^{-1}T^{*}$

$=T\{f(l)\}^{-}1T*$

.

Hence $f(l)$ is increasing for $l\geq k>0$. Moreover,

$( \tau^{*}|\tau|2\iota\tau)\frac{1}{l+1}=f(l)\geq f(k)\geq|T|^{2}$,

that is, $T$ is class $\mathrm{A}(l)$ for $l\geq k>0$. Whence the proof of Theor$e\mathrm{m}3$ is complete.

$\square$

Proof

of

Theorem

_4.

Suppose that $T$ is absolute-k-paranormal for $k>0$, i.e.,

$|||\tau|^{k}\tau_{x}||\geq||TX||^{k}+1$ for every unit vector $x\in H.$ (3.2)

(3.2) holds if and only if

$F(k)=|||T|^{k}Tx||^{\frac{1}{k+1}}\geq||TX||$ for every unit vector $x\in H$

.

Then for every unit vector $x\in H$ and any $l$ such that $l\geq k>0$, we have

$F(l)=|||T|^{l} \tau x||\frac{1}{l+1}$

$=(|T|2lT_{X,T)^{\frac{1}{2(l+1)}}}\backslash x$

$\geq \mathrm{t}(|T|^{2k}\tau X, Tx)^{\frac{l}{k}}||TX||2(1-\frac{l}{k})\}^{\frac{1}{2(l+1)}}$ by (ii) of Theorem B.2

$= \{|||\tau|^{k}\tau_{X1}|\frac{2l}{k}||TX||2\mathrm{t}1-\frac{\iota}{k})\}^{\frac{1}{2\mathrm{t}^{\mathrm{t}+}1)}}$

$\geq\{||Tx||^{\frac{2l(k+1)}{k}}||TX||2(1-\frac{l}{k})\}^{\frac{1}{2\mathrm{t}^{\iota+}1)}}$ by (3.2)

$=||TX||$

.

Hence

$F(l)=|||T|^{l}\tau_{X1}|^{\frac{1}{l+1}}\geq||TX||$ for every unit vector _{$x\in H$} and $l\geq k$, (3.5)

Next we show that $F(l)$ is increasing for $l\geq k>0$. For every unit vector $x\in H$ and

any $m$ and $l$ such that $m\geq l\geq k>0$,

$F(m)=|||\tau|^{m}Tx||^{\frac{1}{m+1}}$

$=(|T|^{2m}T_{X,T_{X}})^{\frac{1}{2i^{m}+1)}}$

$\geq\{(|T|^{2l}TX, T_{X})^{\frac{m}{\iota}}||TX||2\mathrm{t}1-\frac{m}{l})\}^{\frac{1}{2(m+\iota)}}$ by (ii) of Theorem B.2

$= \{|||T|lTX||^{\frac{2m}{l}}||T_{X1}|2(1-\frac{m}{l})\}^{\frac{1}{2(m+1)}}$

$\geq\{|||T|l\tau X||\frac{2m}{l}|||T|^{\iota}TX||^{\frac{2}{\mathrm{t}+1}(}1-\frac{m}{l})\}^{\frac{1}{2\{m+1\}}}\mathrm{b}\mathrm{y}(3.5)$

$=|||T|^{l}Tx|| \frac{1}{\mathrm{t}+1}$

$=F(l)$

.

Hence $F(l)$ is increasing for $l\geq k>0$. Whence the proof ofTheorem 4 is compl$e\mathrm{t}\mathrm{e}$

.

$\coprod$

4

Examples

In this section, we shall give a characterization of $\mathrm{a}\mathrm{b}_{\mathrm{S}}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{e}- k$-paranormal operators.

And by using this characterization, we shall give several examples showing that inclusion

relations among the classes discussed in this paper are all proper. Ando [3] proved the following Theorem D.1.

Theorem D.l ([3]). An operator $T$ is paranormal

if

and only

if

$T^{*2}\tau^{2}-2\lambda T^{*}T+\lambda^{2}\geq 0$

for

all $\lambda>0$

.

As a generalization of Theorem D.l,we have the following characterization of

absolute-$k$-paranormal operators.

Theorem 5. For each $k>0$, an operator $T$ is $ab_{S}olute- k$-paranormal

if

and only

if

$T^{*}|T|2k\tau-(k+1)\lambda^{k}|T|^{2}+k\lambda^{k+1}\geq 0$

for

all$\lambda>0$

.

(4.1)

In fact Theorem 5 implies Theorem D.l putting $k=1$ _in $\mathrm{T}\mathrm{h}_{c}\mathrm{e}$orem 5.

We cite the following well-known lemma in order to give a proof of Theorem 5.

Lemma D.2. Let $a$ and $b$ be positive real numbers. Then

Proof

of

Theorem 5. Suppose that $T$ is absolute-k-paranormal for $k>0$, i.e.,

$|||T|^{k}\tau_{X}||\geq||TX||k+1$ for every unit vector $x\in H.$ (3.2)

(3.2) holds iff

$|||T|^{k}T_{X1}| \frac{1}{k+1}||x||\frac{k}{k+1}\geq||TX||$ for all _{$x\in H$}

iff

$(T^{*}|T|^{2}kTX, x) \frac{1}{k+1}(x, X)^{\frac{k}{k+1}}\geq(|T|^{2}X, x)$ for all _{$x\in H$}. (4.2)

By Lemma D.2,

$( \tau*|\tau|^{2}kTx, X)^{\frac{1}{k+1}(x,X})^{\frac{k}{k+1}}=\{(\frac{1}{\lambda})^{k}(T^{*}|\tau|^{2}k\tau_{Xx},)\}\frac{1}{k+1}\cdot\{\lambda(X, X)\}^{\frac{k}{k+1}}$

$\leq\frac{1}{k+1}\frac{1}{\lambda^{k}}(\tau*|\tau|^{2}k\tau_{x,X})+\frac{k}{k+1}\lambda(x,x)$

(4.3) for all $x\in H$ and all $\lambda>0$,

so (4.2) ensures the following (4.4) by (4.3).

$\frac{1}{k+1}\frac{1}{\lambda^{k}}(\tau^{*}|\tau|^{2}k\tau_{x,x)}+\frac{k}{k+1}\lambda(x, x)\geq(|T|^{2}x, x)$

(4.4) for all $x\in H$ and all $\lambda>0$.

Convers$e1\mathrm{y},$ $(4.2)$ follows from (4.4) by putting $\lambda--\{\frac{(T^{*}|\tau|^{2}kTx,x)}{(x,x)}\}^{\frac{1}{k+1}}$ (In case

$(T^{*}|\tau|^{2}kTx, x)=0$, let $\lambdaarrow 0.$) Hence (4.2) is equivalent to (4.4), and (4.4) holds if and

only if

$T^{*}|T|2k\tau-(k+1)\lambda^{k}|T|^{2}+k\lambda^{k+1}\geq 0$ for all $\lambda>0$, (4.1)

so that the proofis complete. $\square$

The following Proposition 6 is obtained by easy calculations, so we omit to describe

th$e\mathrm{s}\mathrm{e}$ calculations.

Proposition 6. Let $K= \bigoplus_{n=-\infty}^{\infty}H_{n}$ where $H_{n}\cong H$

.

For given positive operators $A,$$B$ on

$H$

,

define

the operator$T_{A,B}$ on $K$ as

follows:

$T_{A,B}=(.\cdot.\cdot.\cdot$ $B0$ $B0$ $A0$ $A0$ $\prime 0..$ $\cdot..$

).

(4.5)

(i) $T_{A,B}$ is $log$-hyponormal

if

and only

if

$A$ and $B$ are invertible and $\log A\geq\log B$

.

(ii) For each $k>0,$ $T_{A,B}$ is class $A(k)$

if

and only

if

$(BA^{2k}B)^{\frac{1}{k+1}}\geq B2$

.

(iii) For each $k>0_{f}T_{A,B}$ is $ab_{S}olute-k$-paranormal

if

and only

if

$BA^{2k}B-(k+1)\lambda^{k}B^{2}+k\lambda^{k+1}\geq 0$

for

all $\lambda>0$

.

By using Proposition 6, we can give several examples to show that inclusion relations

among thes$e$ classes are all proper.

$\mathrm{E}\mathrm{x}\mathrm{a}\mathrm{m}$

.ple

1. Let $K= \bigoplus_{n=-\infty}^{\infty}H_{n}$ where $H_{n}\cong \mathbb{R}^{2}$

.

For given positive matrices $A,$$B$ on

$\mathbb{R}^{2}$,

define

the operator $T_{A,B}$ on $K$ as (4.5) in

$\dot{P}$

roposition 6. Then we have the foflowing

examples.

We remark that the trace of a matrix$X$ denotes tr$X$ and the determinant of a matrix

$X$ denotes $\det X$.

(1) An example

_of

non-log-hyponormal, class $A$ operator.

Let

$A=$

and

$B=$

Fujii, Furuta and Wang [7] shows that $\log A\not\geq\log B$and$A^{2}\geq(AB^{2}A)^{\frac{1}{2}}$ hold together. On

the other hand, $A^{2}\geq(AB^{2}A)^{\frac{1}{2}}$ holds iff $(BA^{2}B)^{\frac{1}{2}}\geq B^{2}$ holds by Lemma B.3. Therefore

$T_{A,B}$ is $\mathrm{n}\mathrm{o}\mathrm{n}-\log-\mathrm{h}\mathrm{y}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}$ but class A by (i) and (ii) of Proposition 6.

(2) An example

_of

non-class $A$, class $A(\mathit{2})$, paranormal operator.

Let

$A=(_{0}^{2}2\sqrt{23}0)$ and

$B=$

.

Then

$(BA^{2}B) \frac{1}{2}-B^{2}=$

.

Eigenvalues of $(BA^{2}B)^{\frac{1}{2}}-B^{2}$ are 12.585... and-0.64001... , so that $(BA^{2}B)^{\frac{1}{2}}\not\geq B^{2}$

.

On the other hand,

$(BA^{4}B)^{\frac{1}{3}}-B^{2}=$

.

Eigenvalues of $(BA^{4}B)^{\frac{1}{3}}-B^{2}$ are24.760.

..

and 0.31608... , so that $(BA^{4}B)^{\frac{1}{3}}\geq B^{2}$

.

So

$T_{A,B}$ is class $\mathrm{A}(2)$ by (ii) ofProposition 6.

Furthermore, for $\lambda>_{\iota}0$, define $X_{1}(\lambda)$ as follows:

$X_{1}(\lambda)=BA^{2}B-2\lambda B^{2}+\lambda^{2}=$

.

Put $p_{1}(\lambda)=\mathrm{t}\mathrm{r}X_{1}(\lambda)$ and $q_{1}(\lambda)=\det X_{1}(\lambda)$, then $p_{1}(\lambda)=2\lambda 2-52\lambda+1^{\cdot}248$ $=2(\lambda-13)^{2}+910>0$ and $q_{1}(\lambda)=(404-26\lambda+\lambda^{2})(844-26\lambda+\lambda^{2})-(-576+24\lambda)^{2}$ $=\lambda^{4}-52\lambda^{3}+1348\lambda^{2}-48\mathrm{o}\mathrm{o}\lambda+9200$

.

By calculation, $q_{1}’(\lambda)=4\lambda^{3}-156\lambda^{2}+2696\lambda-4800$ $=4(\lambda-2)(\lambda^{2}-37\lambda+600)$ $=4( \lambda-2)\{(\lambda-\frac{37}{2})^{2}+\frac{1031}{4}\}$ .

So $q_{1}’(\lambda)=0$ iff $\lambda=2$, that is, $q_{1}(\lambda)\geq q_{1}(2)=4592>0$ for all $\lambda>0$

.

Hence $X_{1}(\lambda)\geq 0$

for all $\lambda>0$ since tr$X_{1}(\lambda)=p_{1}(\lambda)>0$ and$\det x_{1}(\lambda)=q_{1}(\lambda)>0$ forall $\lambda>0$

.

Therefore

$T_{A,B}$ is paranormal by (iii) ofProposition 6.

(3) An example

_of

non-class $A(\mathit{2}),$ $ab_{S}olute- \mathit{2}$-paranormal operator.

Let

$A=$

and $B= \frac{1}{2}(_{1-\sqrt{3}}^{1+\sqrt{3}}1+\sqrt{3}1-\vee 3\gamma$

.

Then

$(BA^{4}B) \frac{1}{3}-B^{2}=$

.

Eigenvalues of $(BA^{4}B)^{\frac{1}{3}}-B^{2}$ are

1.4151...

and-0.14687.

..

,

so that $(BA^{4}B)^{\frac{1}{s}}\not\geq B^{2}$

.

On the other hand, for $\lambda>0$, define $X_{2}(\lambda)$ as follows:

$X_{2}(\lambda)=BA4B-3\lambda 2B2+2\lambda 3=(^{24-8\sqrt{3}-6\lambda}-12+3\lambda 2^{+2\lambda}2324+8\sqrt{3}-6\lambda 2+2\lambda^{3)}-12+3\lambda 2$

.

Put $p_{2}(\lambda)=\mathrm{t}\mathrm{r}X_{2}(\lambda)$ and $q_{2}(\lambda)=\det x_{2}(\lambda)$, then

$p_{2}(\lambda)=4\lambda^{\mathrm{s}_{-1}2}2\lambda+48$

and

$q_{2}(\lambda)=(24-8\sqrt{3}-6\lambda^{2}+2\lambda^{3})(24+8\sqrt{3}-6\lambda^{2}+2\lambda^{3})-64$

$=4\lambda^{6}-24\lambda^{5}+27\lambda^{4}+96\lambda^{3}-216\lambda^{2}+240$

.

Weeasily obtain $p_{2}(\lambda)>0$ for all $\lambda>0$

.

And we have

$q_{2}’(\lambda)=24\lambda^{5}-120\lambda^{4}+108\lambda^{3}+288\lambda^{2}-432\lambda$

$=12\lambda(\lambda-2)(2\lambda^{3}-6\lambda 2-3\lambda+18)$

.

So $q_{2}’(\lambda)=0$ iff$\lambda=0,2$ since$2\lambda^{3}-6\lambda^{2}-3\lambda+18>0$ for all $\lambda>0$ by an $\mathrm{e}\mathrm{a}\dot{\mathrm{s}}\mathrm{y}$calculation,

that is, $q_{2}(\lambda)\geq q_{2}(2)=64>0$ for all $\lambda>0$. Hence $X_{2}(\lambda)\geq 0$ for all $\lambda>0$ since

tr$X_{2}(\lambda)=p_{2}(\lambda)>0$ and $\det X_{2}(\lambda)=q_{2}(\lambda)>0$ for all $\lambda>0$

.

Therefore $T_{A,B}$ is

absolute-2-paranormal by (iii) of Proposition 6.

(4) An example

_of

non-paranormal, $ab_{S}olute- \mathit{2}$-paranormal operator.

Let

$A=$

and

$B=$

.

Then for $\lambda>0$, define $X_{3}(\lambda)$ as follows:

$X_{3}(\lambda)=BA^{2}B-2\lambda B^{2}+\lambda^{2}=$

.

(4.6)

Put $\lambda=4$ in (4.6), then

$X_{3}(4)=\not\geq 0$

.

So $T_{A,B}$ is non-paranormal by (iii) of Proposition 6.

On the other hand, for $\lambda>0$, define $X_{4}(\lambda)$ as follows:

$X_{4}(\lambda)=BA^{4}B-3\lambda^{2}B^{2}+2\lambda^{3}=$ .

By an easy calculation, $80-12\lambda^{2}+2\lambda^{3}>0$ for all $\lambda>0$. So $X_{4}(\lambda)\geq 0$ for all

$\lambda>0\square$’

5

Remarks

$\mathrm{r}e$sult, we obtain the following Theoren 7.

Theorem 7.

_If

an operator $T$ is $ab_{SO}lute- k$-paranormal

for

some $k>0$, then $T$ is

nor-maloid.

Proof

of

Theorem 7. In case $T$ is $\mathrm{a}\mathrm{b}\mathrm{S}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}e- k- \mathrm{P}^{\mathrm{a}\mathrm{r}}\mathrm{a}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}1$ for some

$0<k<1,$

$T$ is

paranormalby Theorem 4, so that $T$is normaloid as shown in [9] and [15]. Sowe consider

only the case of$k\geq 1$. Suppose $T$ is absolute-k-paranormal for some $k\geq 1$, i.e.,

$|||\tau|^{k}\tau_{x}||\geq||TX||k+1$ for every unit vector _$x\in.H$

.

(3.2)

(3.2) holds iff

$|||T|^{k}\tau X||||x||^{k}\geq||TX||k+1$ for all $x\in H.$ (5.1)

We shall show that

$||T^{n}||=||T||^{n}$ (5.2)

for all positive integers $n$. When $n=1,$ $(5.2)$ is always holds. Assume that (5.2) $\mathrm{h}\mathrm{o}.1$ds for

, some positive integer $n$

.

Then for

ev.ery

unit vector $x\in H$, wehave

$||T^{n}x||^{k+}1=||T\cdot\tau^{n}-1X||k+1$ $\leq|||T|^{k}\tau\cdot T^{n-1}X||\cdot||\tau^{n-1}x||^{k}$ by (5.1) $\leq|||T|^{k-1}||\cdot|||T|\tau^{n}x||\cdot||T^{n-1}||^{k}$ $\leq||T||^{k-1}\cdot||T^{n+1}X||\cdot||T||^{(n-1})k$ $\leq||T^{n+1}||\cdot||T||^{n}k-1$, that is, $||T^{n+1}||\cdot||T||^{n}k-1\geq||T^{n}||k+1$ (5.3)

By the assumption (5.2) for $n,$ $(5.3)$ holds if and only if $||T^{n+1}||\cdot||T||^{n}k-1\geq||T||^{n}\mathrm{t}k+1)$,

that is $||T^{n+1}||\geq||T||n+1$, so that $||T^{n+1}||=|!^{\tau||^{n+1}}$

Therefore $||T^{n}||=||T||^{n}$ for all positive integers $n$ by induction. Hence the proof

$\mathrm{o}\mathrm{f}\square$

Theorem 7 is complete.

We cangive anexample toshowthat inclusionrelations between absolut$e- k$-paranormal

Example 2. There exists a $non- ab_{S}olute-k$-paranormal

for

any $k>0$ and normaloid operator.

LLet

$T=$

.

Then $||T^{n}||=||T||^{n}$ forallpositiveintegers $n$by aneasy calculation. However,the relation

$|||T|k\tau_{x1}|\geq||TX||k+1$ does not hold for the unit vector $e_{2}=(0,1,0)$ since

$|T|^{k}T=$

.

Hence $T$ is non-absolute-k-paranormalfor any $k>0$

.

$\square$

Remark 2. The following diagram expresses the inclusion relations among the classes

References

[1] A.Aluthge, On$p$-hyponormal operators

for

$0<p<1$ , Integr. Equat. Oper. Th., 13 (1990),

307-315.

[2] A.Aluthge,Some generalizedtheoremson$p$-hyponormaloperators, Integr. Equat. Oper. Th.,

24 (1996), 497-501.

[3] T.Ando, Operators $with$ a norm condition, Acta Sci. Math. Szeged, 33 (1972), 169-178.

[4] T.Ando, On some operator inequalities, Math. Ann., 279 (1987), 157-159.

[5] M.Cho and M.ltoh, Putnam’s inequality

_for

$p$-hyponormal opoerators, Proc. Amer. Math.

Soc., 123 (1995), 2435-2440.

[6] M.Fujii, T.Furuta and E.Kamei, Furuta’s inequality and its application to Ando’s theorem,

Linear Algebra Appl., 179 (1993), 161-169.

[7] M.Fujii, T.Furutaand D.Wang, An application

_of

the Furuta inequality to operator

inequal-ities on chaotic orders, Math. Japon., 40 (1994), 317-321.

[8] M.Fujii, R.Nakamoto and H.Watanabe, The $Heinz-Kato-Fu\Gamma uta$ inequality and hyponormal

operators, Math. Japon., 40 (1994), 469-472.

[9] T.Furuta, On the class

_of

paranormal operators, Proc. Japan Acad., 43 (1967), 594-598. [10] T.Furuta, Applications

_of

orderpreserving operator inequalities, Oper. Theory Adv. Appl.,

59 (1992), 180-190.

[11] T.Furuta, Extension

_of

the Furuta inequality and A$ndo$-Hiai $log$-majorization, Linear

Alge-bra Appl., 219 (1995), 139-155.

[12] T.Furuta, T.Yamazaki and M.Yanagida, Order$prese\Gamma\dot{m}ng$ opemtof

function

via Furuta

in-equality $‘\ell A\geq$ B $\geq$ 0 ensures

$(A^{\frac{r}{2}}A^{p}A \frac{r}{2})^{\frac{1+f}{p+r}}\geq(A^{\frac{f}{2}}B^{p}A\frac{r}{2})^{\frac{1+r}{\mathrm{p}+r}}$

for

p $\geq$ 1 and r $\geq$ 0”, to

appear in Proc. 96-IWOTA.

[13] T.Furuta and M.Yanagida, Further exten8ions

of

Aluthge

transformation

on p-hyponormal

operators, Integr. Equat. Oper. Th., 29 (1997), 122-125.

[14] T.Huruya, A note on $p$-hyponormal operators, Proc. Amer. Math. Soc., 125 (1997),

3617-3624.

[15] $\mathrm{V}.\mathrm{I}\mathrm{s}\mathrm{t}\mathrm{r}\dot{\mathrm{a}}\mathrm{t}\mathrm{e}\mathrm{S}\mathrm{C}\mathrm{u}$, T.Saito and T.Yoshino, On a class ofopaerators, Tohoku Math. J., 18 (1966),

410-413.

[16] $\mathrm{C}.\mathrm{A}.\mathrm{M}_{\mathrm{C}}\mathrm{C}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{y},$

$c_{p}$, Israel J. Math., 5 (1967), 249-271.

[17] K.Tanahashi, On $log$-hyponormal operators, preprint.

[18] T.Yoshino, The$p$-hyponormality

of

the Aluthge transform, Interdiscip. Inform.Sci., 3 (1997),

91-93. ’